True or False: The tangent line to a curve can never intersect a slant asymptote to that curve...............

Answer 1

false


Wait, what do you mean by "THE" tangent line?


Consider f(x)=(x^3-1)/(x^2+2). Then f has a slant asymptote of g(x)=x.

But f'(x)= [(x^2+2)(3x^2) - (x^3-1)(2x)] /[(x^2+2)^2]

f'(2)= [(4+2)(12)-(8-1)(4)]/(36) = (72-28)/36= 44/36>1

since f(2)=7/6<2, then the tangent line to f at x=2 must cross the line g (which is the slant asymptote).

Answer 2

False! Consider the function y = SQRT(1+x^2). (This is the top branch of the hyperbola y^2-x^2 = 1.) The tangent line at x=0 is the horizontal line y=1. The slant asymptotes to the function are y = x (as x-> infinity) and y = -x (as x->minus infinity).

These slant asymptotes clearly intersect the line y=1.

Answer 3

False: if it were true, then the curve would always have to be parallel to its slant asymtote. Otherwise, the tanget line would not be parallel, and thus intersect, according to the laws of Geometry. However, you said it was a curve and not a straight line, so it can't always be parallel to its slant asymtote because it's a curve. So no, false.

Answer 4

The function f(x) = x + 1/x^2 has y = x as an asyptote, yet the tangent to this curve at x = 1, which is y = 3 - x, intersects y = x at (1.5, 1.5).

So, FALSE!

Answer 5

False.

Answer 6

false

Answer 7

seems true.